Determination of Mixed-mode Stress Intensity Factors, Fracture

7/26/2019 Determination of Mixed-mode Stress Intensity Factors, Fracture

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Determination of mixed-mode stress intensity factors, fracture

toughness, and crack turning angle for anisotropic foam material

Nagaraj K. Arakere a,*, Erik C. Knudsen a, Doug Wells b, Preston McGill b, Gregory R. Swanson b

a Mechanical & Aerospace Engineering Department, University of Florida, 237 MAE-B, Gainesville, FL 32611-6300, United Statesb NASA Marshall Space Flight Center, Huntsville, AL-35812, United States

a r t i c l e i n f o

Article history:

Received 6 July 2007

Received in revised form 19 March 2008

Available online 11 May 2008

Keywords:

Foam

Anisotropy

Stress intensity factor

Mixed-mode loading

Fracture toughness

Crack turning angle

Anisotropic toughness

a b s t r a c t

A numerical and experimental investigation for determining mixed-mode stress intensity

factors, fracture toughness, and crack turning angle for BX-265 foam insulation material,

used by NASA to insulate the external tank (ET) for the space shuttle, is presented.

BX-265 foam is a type of spray-on foam insulation (SOFI), similar to the material used to

insulate attics in residential construction. This cellular material is a good insulator and is

very lightweight. Breakup of segments of this foam insulation on the shuttle ET impacting

the shuttle thermal protection tiles during liftoff is believed to have caused the space shut-

tle Columbia failure during re-entry. NASA engineers are interested in understanding the

processes that govern the breakup/fracture of this material from the shuttle ET. The foam

is anisotropic in nature and the required stress and fracture mechanics analysis must

include the effects of the direction dependence on material properties. Material testing

at NASA Marshall Space Flight Center (MSFC) has indicated that the foam can be modeled

as a transversely isotropic material. As a first step toward understanding the fracture

mechanics of this material, we present a general theoretical and numerical framework

for computing stress intensity factors (SIFs), under mixed-mode loading conditions, taking

into account the material anisotropy. We present SIFs for middle tension M(T) test

specimens, using 3D finite element stress analysis (ANSYS) and FRANC3D fracture analysis

software. SIF values are presented for a range of foam material orientations. Mode I fracture

toughness of the material is determined based on the SIF value at failure load. We also

present crack turning angles for anisotropic foam material under mixed-mode loading.

The results represent a quantitative basis for evaluating the strength and fracture proper-

ties of anisotropic foam insulation material.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

On February 1, 2003 the Space Shuttle Columbia suffered a catastrophic failure during re-entry. NASA has conducted an

exhaustive investigation of the failure and the consensus now is that the breakup was caused by a segment of foam insula-

tion, roughly the size of a suitcase, striking the wing during liftoff. The foam impact, in prior shuttle launches, has been

known to cause impact damage to the thermal protection tiles, but was not considered to be a serious problem until the

Columbia disaster. The damage to the thermal protection tiles in the leading edge of the wing is thought to have initiated

thermal damage during re-entry, triggering a cascading series of catastrophic events that led to the loss of the shuttle.

The foam insulation is sprayed in layers on the cylindrical ET surface in liquid form which then expands and rises in the

normal direction and a rind or knit line forms when the foam has settled and stops expanding. Referring to Fig. 1, the rise

0020-7683/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2008.04.028

* Corresponding author. Tel.: +1 352 392 0856: fax: +1 352 392 1071.

E-mail address: [email protected](N.K. Arakere).

International Journal of Solids and Structures 45 (2008) 49364951

Contents lists available at ScienceDirect

International Journal of Solids and Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c at e / i j s o l s t r
mailto:[email protected]://www.sciencedirect.com/science/journal/00207683http://www.elsevier.com/locate/ijsolstrhttp://www.elsevier.com/locate/ijsolstrhttp://www.sciencedirect.com/science/journal/00207683mailto:[email protected]


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direction is denoted by the 33 direction. The knit lines are also shown in Fig. 1. A second layer is then sprayed on and allowed

to expand and the process continued until the insulation layer reaches the required thickness. Foam is a cellular material

comprised of many individual cells at the microscopic level. During the spraying process, the sizes of the cells typically vary

in the rise, axial and circumferential (hoop) directions, making the material properties direction dependent.

Certain areas of the ET have geometric discontinuities such as bolts, flanges, and fittings, as shown in Fig. 2. When the

foam insulation is sprayed onto these areas voids can be created because of the geometric discontinuities. Furthermore,

as the foam expands in these regions of, the rise directions normal to bolts and flanges will likely cause different material

orientations and hence different anisotropic properties compared to areas that are relatively flat. The tank is filled with liquid

oxygen and hydrogen and therefore the foam is exposed to cryogenic temperatures at the ETs surface. Liquid nitrogen and

oxygen (air) can condense into any voids present. During liftoff the outer surface of the foam is exposed to aerodynamic

heating. This heating is thought to raise the temperature of the liquid nitrogen, turning it into a gas. The pressure difference

associated with the formation of the gas can cause pieces of foam to be blown out during liftoff.

NASA MSFC has undertaken extensive testing to measure the elastic and fracture properties of the BX-265 foam that

insulates the ET. Fig. 3 shows the middle tension M (T) fracture specimen used for evaluating fracture toughness.

Fig. 4 shows the tensile test specimens used for determining elastic constants. NASA is performing an investigation to

study the effects of material orientation on strength and fracture properties of the foam material. Evaluating the varia-

tion of SIF, fracture toughness, and crack turning angle, as a function of material orientation is a first step towards a

broader understanding of the failure processes of the foam material at certain critical areas of the shuttle ET containing

bolts, flanges, and fittings.

A wide range of bending, shear, and tensile tests were performed at NASA MSFC, to obtain the elastic properties of foam,

shown in Table 1. The tensile loaddeflection behavior showed that the foam behaves as a brittle material and fractures with

small deformation. Although foam is a cellular material comprised of many individual cells at the microscopic level, for the

purposes of this initial investigation, we will assume that the material behaves as a homogeneous anisotropic elastic solid

and follows laws of linear elastic fracture mechanics. Based on elastic properties shown in Table 1 this material can be

approximated as a transversely isotropic material, with the plane of isotropy being 1122. The material coordinate system

used to define the transverse isotropy is shown inFig. 5. The foam material is stiffer in the rise direction than the axial and

hoop directions.

The objectives for this paper are to present a general numerical and experimental procedure for evaluating mixed-mode

SIF (KI,KII,KIII), fracture toughness for a M(T) foam specimen shown inFig. 3, and crack turning angle, as a function of mate-

rial orientation, considering the foam as a transversely isotropic material.

2. Computation of stress intensity factors for cracks in anisotropic materials

The computation of SIFs for cracks subjected to mixed-mode loading in isotropic elastic solids can be accomplished in a

straightforward manner using a number of analytical and numerical methods. Aliabadi and Rooke (1991)present a detailed

summary of available methods for analytical and numerical evaluation of SIFs in isotropic materials. Some commonly used

Fig. 1. Material coordinate system for spray-on foam insulation (SOFI).

N.K. Arakere et al. / International Journal of Solids and Structures 45 (2008) 49364951 4937


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Fig. 2. Areas on the shuttle ET, with geometric discontinuities, that are prone for loss of foam insulation material during liftoff.

Fig. 3. Middle tension, M(T), fracture test specimen used at NASA MSFC.

Fig. 4. Foam tensile test specimens used for determining elastic constants.

4938 N.K. Arakere et al./ International Journal of Solids and Structures 45 (2008) 49364951


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methods are the J-integral approach (Rice, 1968), virtual crack extension (Parks, 1974; Ishikawa, 1980), modified crack clo-

sure integral and displacement extrapolation methods (Raju, 1987). However, none of these methods can be used for obtain-

ingK-solutions for a crack in a general anisotropic solid subject to mixed-mode loading (see Fig. 6).

For a crack front subjected to mixed-mode loading, near-tip displacement fields are a linear combination of the three

deformation modes. For general anisotropy there is full coupling between KI, KII, KIII and the three modes of deformation.

Forz-symmetry (plane of material symmetry is perpendicular to the crack front), there is decoupling between the in plane

(xyplane (KI,KII)) and out of plane deformations (KIII). For isotropy there is complete decoupling between the three defor-

mation modes with respect to the K-solutions. Ranjan and Arakere (2008) have presented mixed-mode SIFs for az-symmetry

crack in a FCC single crystal material. FCC single crystals have cubic symmetry requiring three independent elastic constants

to relate stress and strain.

Sih et al. (1965)presented the stress and displacement fields near a crack tip for the case ofz-symmetry.Hoenig (1982)

presented the stress and displacement fields near a crack tip for general anisotropy. The concept of a M-integral, M(1,2) (Yau

et al., 1980) is based on the linear superposition of two separate and independent equilibrium states. Banks-Sills et al. (2007)

draw upon the work ofSih et al. (1965), Hoenig (1982), Yau et al. (1980), Li et al. (1985) andBanks-Sills et al. (2005)and

present a comprehensive numerical framework for computing SIFs for cracks in anisotropic materials, which form the basis

for the current FRANC3D FEA based fracture analysis code developed by the Cornell University fracture group ( FRANC3D

Concepts and Users Guide, 2003). Earlier versions of FRANC2D and FRANC3D, based on work by Swenson and Ingraffea

(1987, 1988a,b), were primarily used for isotropic materials. With an adaptive mesh technique, this fracture software can

simulate crack growth without prescribing the crack path. Their work has provided fundamental understandings for the sim-

ulation of dynamic crack propagation based on FEA. We present a brief synopsis of the pertinent equations necessary for

evaluating SIFs under mixed-mode loading for anisotropic materials, based on Banks-Sills et al. (2007). RicesJ-integral (Rice,

1968) represents the energy flux into the crack tip region, and is equal to the energy release rate under conditions of small-

scale yielding. The three-dimensional form of the J-integral was first presented byShih et al. (1986)and later re-derived by

Table 1

Experimentally measured material elastic constants

E11= 950 psi (6.55MPa) m12= 0.45 G12= 328 psi (2.26 MPa)E22= 950 psi (6.55MPa) m23= 0.3 G23= 231 psi (1.59 MPa)E33= 2400 psi (16.54 MPa) m13= 0.3 G31= 231 psi (1.59 MPa)

33

22

11

Fig. 5. Coordinate system used for the transversely isotropic foam material, with 33 as the foam rise direction.

Fig. 6. Coordinate system used for the local crack front.



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Freed and Banks-Sills (2005), and is reproduced here as

Z LN0

GzlNx znxdzZ

V

rijouiox1

Wd1j

oq1oxj

dV 1

where G is the energy release rate along the crack front, dllNx nx is the normalized virtual crack extension orthogonal to thecrack front,nx is the unit normal to the crack front in the x-direction,Nrepresents the element number Nalong the crack

front, andLNis its length, W= rijeij/2 is the strain energy density, and d ij is the Kronecker delta. The subscripts i and j areused to represent crack tip coordinatesx,y andz, andrij,eijanduirepresent the stress, strain and displacement componentsin the crack tip coordinate system. This form of theJ-integral was utilized to derive a three-dimensional form of theM-inte-

gral (Banks-Sills et al. (2007)) for the general anisotropic case, as

M1;2a 1Aq

ZV

r1ijou

2ai

ox1 r2aij

ou1i

ox1W1;2ad1j

" #oq1oxj

dV 2

wherea = a,b andc in succession, the area of virtual crack extension is given by

AqZ LN

0

lNx zdz 3

and the interaction energy density

W1;2ar1ij e2aij r2aij e1ij 4The M-integral describes the interaction between the two equilibrium states, denoted by the superscripts 1 and 2, which can

be superposed since the material is linearly elastic. The functionq is one at the crack tip and zero on the arbitrary boundary

surface S that defines the volume Vof the integration domain, and can be interpreted as a virtual crack extension (Freed and

Banks-Sills (2005)). The relationship between the average value ofMthrough the element thickness, denoted byM, and the

stress intensity factors for a straight crack in a general anisotropic solid for which the material and crack coordinates may be

at arbitrary angles, is given below (Banks-Sills et al., 2007)

M1;2a 12Aq

2K1I Im m2iN

1i1

K1II Im m1iN1i1 m2iN1i2

K1III Im m3iN1i1 m2iN1i3

n o 5

M1;2b 12Aq

K1I Im m2iN

1i2 m1iN1i1 2K

1II Im m1iN

1i2 K

1IIIIm m3iN

1i2 m1iN1i3 n o 6

M1;2c 12Aq

K1I Im m2iN

1i3 m3iN1i1

K1II Im m1iN1i3 m3iN1i2

2K1IIIIm m3iN1i3

n o 7

Eqs.(5)(7), expressed in matrix form, lead to the relationship between stress intensity factors and M, shown below:

2Im m2iN1i1

Im m1iN

1i1 m2iN1i2

Im m3iN

1i1 m2iN1i3

Im m2iN

1i2 m1iN1i1

2Im m1iN

1i2

Im m3iN

1i2 m1iN1i3

Im m2iN

1i3 m3iN1i1

Im m1iN

1i3 m3iN1i2

2Im m3iN

1i3

266664

377775

K1I

K1II

K1III

8>>>:

9>>=>>; 2Aq

M1;2a

M1;2b

M1;2c

8>:

9>=>; 8

The parametersmij are given by

m1i

S011l2i

S016li

S012

ki S

015li

S014

m2iS021liS026S022=liki S025S024=li

m3iS041liS046S042=liki S045S044=li 9

where, under plane strain condition S0ijSij Si3S3j=S33, andSij (i,j= 1,. . ., 6) is the contracted material compliance matrixobtained from the fourth order tensor Sijkl (i,j,k,l = 1, 2, 3) andeij= Sijklrkl (Ting, 1996).

Theparametersli (i = 1,2, 3)arethe rootsofthesixthordercharacteristicequationof theplanestraincompliancematrixgivenas

l4ll2l l23l 0 10where

l2l S055l2 2S045lS044l3l S015l3 S014S056

l2 S025S046 lS024

l4l S011l4

2S016l3

2S012S066 l2 2S026lS022



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The rootsli(i= 1, 2, 3) occur in complex conjugate pairs and the three roots with positive imaginary parts are selected. Thematrix Nij (singular for isotropy and z-symmetry) is given by

Nij1 1 1

l1 l2 l3k1 k2 k3

264

375 11

The parameters ki are given by

ki l3lil2li

; i1; 2; 3 12

A transversely isotropic linear elastic solid has 5 independent elastic constants given in Table 1. The relationship between the

constants, with the plane of isotropy being 1122 (Fig. 5), are E11= E22, G23= G13,m12= m21,G12= E11/{2(1 + m12)} andm ij

Ej mji

Ei

(summation not implied). The strainstress relations are given as

e11

e22

e33

c23c13

c12

8>>>>>>>>>>>>>>>:

9>>>>>>>>=

>>>>>>>>;

1E1

m12E2

m13E3

0 0 0

m21E1

1E2

m23E3

0 0 0

m31E1

m32E2

1E3

0 0 0

0 0 0 1G23

0 0

0 0 0 0 1G13

0

0 0 0 0 0 1G12

266666666664

377777777775

r11r22r33r23r13

r12

8>>>>>>>>>>>>>>>:

9>>>>>>>>=

>>>>>>>>;13

Eq. (8) provides the basis for computing SIFs, provided that a numerical procedure is implemented for computing the M-inte-

grals in the RHS vector. The fracture code FRANC3D has been set up to take the displacement and stress field input from a

anisotropic FEA crack mesh model using commercial software such as ANSYS or ABAQUS, compute the elements of the ma-

trix andM-integral RHS vector in Eq. (8), and solve for the Kvalues along the crack front. This entails the following steps

(Knudsen, 2006; Knudsen and Arakere, 2006): (a) computation of material compliance and stiffness matrices and rotating

them into the global model coordinate system, (b) determination of a local crack front coordinate system and translation

and rotation of all crack front coordinates into the local system, (c) determination of the material class in the rotated con-

figuration, i.e. isotropy,z-symmetry or general anisotropy, (d) computation ofliforz-symmetry or li, ki, mij, and N1 for gen-

eral anisotropy, (e) computation of auxiliary displacements at the crack front using analytical expressions, (f) computation of

displacement derivatives at the integration points, (g) computation of stresses at the integration points from the analytical

expressions, (i) integrate numerically to determine the M-integrals, (j) compute the area of the virtual crack extension, (k)

solve Eq.(8)for the unknown stress intensity factors.

3. Finite element analysis

A middle tension M(T) specimen loaded in tension was modeled using ANSYS commercial finite element software (ANSYS

Elements Reference, 1999), shown in Fig. 7. The specimen is 12 in. (305 mm) long, 5.5 in. (140 mm) wide and 1.5 in. (38 mm)

thick. The center crack in the foam is cut with a razor blade. The 3D model is comprised entirely of SOLID92 and SOLID95

elements with the ability to account for material anisotropy. A 1-psi (6895 Pa) tensile normal stress was applied to the

top face of the model. This results in an 8.25 lb (36.7 N) tensile load being applied to the specimen, since the area of cross

section of the specimen is 8.25 in2 (0.00532 m2). We compute the SIFs for this load and since SIF scales linearly with load, we

can readily estimate Kvalues for other loads. Based on the fracture load for the test specimen, we can then estimate the frac-

ture toughness for the material.

4. Material orientation

There is potential for variation of material orientation relative to the ET, in regions of geometric discontinuities such as

bolts and flanges since the foam rise direction will not be normal to the ET surface, as shown inFig. 8. To perform a system-

atic study of variation of material orientation we adopt a system illustrated in Fig. 9. The inner cone with the yellow dots has

an included angle of 60and the outer cone with the blue dots has an included angle of 90. The numbered dots indicate case

numbers, 0-8 and 916, and represent specific material orientations. Case 0 corresponds to theZ-axis with no tilt, i.e. there is

no relative offset between the substrate and the material coordinate system. For each case the Z0-axis passes through thatnumbered dot and this can be achieved by appropriate rotations about the XandY0 axes. For example, case 1, shown by(X0Y0Z0) in Fig. 9, represents the material orientation with the Z0 axis passing through the point 1 of the inner cone, obtainedby a positive rotation about the X-axis by 30. Similarly, material orientation for case 2 is obtained by a positive rotation

about the X-axis by 22.2 and a (negative) rotation about the Y0-axis by22.2. The required rotations about the Xand Y0axes for all 17 cases are listed in Table 2. The 17 material orientations analyzed are considered to encompass most variations

encountered during the spraying of all parts of the ET.



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Fig. 7. Middle tension M(T) test specimen model built in ANSYS.

Fig. 8. Variation of material orientation relative to the ET.

45

9

5

Z Z

Y

X

321

4

45

867

10 11

12

139

141516

30

13

7

68

5

42

1

3

0

10

11

12

14

15

16

X

Y

X

Y

30

Fig. 9. Specific foam material orientations selected for analysis. The (X0Y0Z0) material orientation represents case 1, with theZ0 axis tilting 30to theZ-axis.(For interpretation of the references to color in this figure citation, the reader is referred to the web version of the article.)



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The material orientation for case 1 is further explained in Fig. 10. The local or primed coordinate system is obtained by apositive 30rotation about theX-axis. This corresponds to the knit lines in the foam oriented at a 30angle to they-axis. The

direction cosines to the primed system for cases 016, required for stress transformations between the primed and unprimed

systems, can be readily obtained using the rotations listed in Table 2.

5. Stress intensity factor results

Finite element models were run in ANSYS for the 17 material orientations defined inFig. 9, using the material properties

defined inTable 1, and the displacement and stress field results input to FRANC3D for post-processing and computation of

SIFs. Crack inclination angle to the horizontal, /, of 0 and 30 were used. Stress intensity factors are presented as a function of

normalized crack front distance or distance through the thickness, with zero indicating the front face and one indicating

the specimen back face. Fig. 11shows the mode I SIF for zero degree crack inclination, for cases 0 and 14, for a far-field

loadingr = 1 psi (6895 Pa). Since the foam material is a lightweight and low strength material, a very low far field loadingofr = 1 psi (6895 Pa) is used. Again, this corresponds to an 8.25 lb (36.7 N) tensile load applied to the 12 in. (305 mm) long,5.5 in. (140 mm) wide and 1.5 in. (38 mm) thick test specimen with crack lengtha = 0.0254 m (1 in.). Cases 0, 1 and 3 have

symmetricKIvalues as a function of thickness, while cases 2 and 4 are anti symmetric with respect to each other. The sym-

metric cases have a maximumKIvalue at the specimen mid plane. KIvalues at the specimen mid plane inFig. 11vary from

2 psiffiffiffiffiffiffi

in:p

2198 Pa ffiffiffiffiffimp to 2:25 psi ffiffiffiffiffiffiin:p 2473 Pa ffiffiffiffiffimp . Fig. 12shows KI results for mode I SIF for a 30 crack inclination, forcases 0 and 14.Figs. 13 and 14 showKIIandKIIIvalues for a 30 crack inclination, for cases 0 and 14.

The mid-plane KIvalues for zero degree crack inclination, shown in Table 2, are examined in further detail, for all 17 cases

analyzed. Case 0 has a minimumKI value of 1:99 psiffiffiffiffiffiffi

in:p

2187 Pa ffiffiffiffiffimp . Cases 3 and 7 (for the 30cone) and correspondingcases 11 and 15 (for the 45 cone) exhibit the maximum KI values of 2:25 psi

ffiffiffiffiffiffiin:

p 2473 Pa ffiffiffiffiffimp and

2:27 psiffiffiffiffiffiffi

in:p

2495 Pa ffiffiffiffiffimp , respectively. This represents a 14% difference between the minimum (case 0) and maximum

Table 2

Specimen mid plane KI values for crack inclination / = 0

Case number KIPaffiffiffiffiffi

mp KIpsi


p X-axis rotation () Y0-axis rotation ()

0 2187 1.99 0 0

1 2209 2.01 30 0

2 2352 2.14 22.2 22.23 2473 2.25 0 304 2352 2.14 22.2 22.25 2209 2.01 30 06 2352 2.14 22.2 22.27 2473 2.25 0 30

8 2352 2.14 22.2 22.2

9 2275 2.07 45 0

10 2407 2.19 35.3 35.311 2495 2.27 0 4512 2407 2.19 35.3 35.313 2275 2.07 45 014 2407 2.19 35.3 35.315 2495 2.27 0 45

16 2407 2.19 35.3 35.3

MeanKI 2360 Paffiffiffiffiffi

mp 2:148 psi


p [a= 0.254 m (1 in.),r = 6895Pa (1 psi), 1 psi


p 1099 Pa ffiffiffiffiffimp ].

Fig. 10. Definition of case one material orientation for the M(T) FE model.



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(cases 11 and 15, 45inclination in rise direction) KIvalue, indicating that the KIvalues are not very sensitive to change in

material orientation.

6. Anisotropic fracture toughness

Fracture toughness can be estimated based on the failure load of M(T) specimen and scaling the KIvalue linearly with ap-

plied load. The failure load for 3 M(T) test specimens with a zero degree crack inclination, and approximately case 0 orienta-

tion are85 lb(378 N), 95lb (422 N), and 106 lb (471 N). The KI value for case 0 with an applied tensile load of 8.25 lb (36.7 N)

is 1:99 psiffiffiffiffiffiffi

in:p

2187 Pa ffiffiffiffiffimp . Scaling linearly with load, this yields fracture toughness values of 858:25

1:9920:5 psi


p 22; 530 Pa ffiffiffiffiffimp , 95

8:25 1:9922:9 psi


p 25; 167 Pa ffiffiffiffiffimp , and 106

8:25 1:9925:5 psi


p 28; 025 Pa ffiffiffiffiffimp , respec-

tively. The mean fracture toughness value from the three data points is 23 psiffiffiffiffiffiffi

in:p

25; 277 Pa ffiffiffiffiffimp . Using the meanKIvaluefrom Table 22360 Pa ffiffiffiffiffimp 2:148 psi ffiffiffiffiffiffiin:p as a scaling factor to account for variation in material orientation, instead of1:99 psi


p 2187 Pa ffiffiffiffiffimp for case 0 orientation alone yields a mean fracture toughness of 2:148

1:99 2324:8 psi


p

27; 255 Pa ffiffiffiffiffimp . From these arguments, a mode I plane strain fracture toughness value in the rise (33) direction between23 and 25 psi


p 25; 277 27; 475 Pa

ffiffiffiffiffim

p is a reasonable figure for the BX-265 foam material. Based on additionaltesting, the fracture toughness in the 11 and 22 directions is estimated to be 17

:4 psi ffiffiffiffiffiffiin

:

p

19;

122 Pa ffiffiffiffiffimp and19:5 psi ffiffiffiffiffiffiin:p 21; 430 Pa ffiffiffiffiffimp . It is worth noting that the units are inpsi ffiffiffiffiffiffiin:p Pa ffiffiffiffiffimp andnotksi ffiffiffiffiffiffiin:p MPa ffiffiffiffiffimp .In general, materials may exhibit elastic anisotropy as well as anisotropy in fracture resistance. Many materials are vir-

tually isotropic elastically, but have a preferred direction of (mode I) crack propagation resulting from the manner in which

the material is processed. If the processing is of symmetric character, as for rolled sheet or plate, the two-dimensional rela-

tion describing the crack growth resistance as a function of orientation has two axes of symmetry. For convenience, the ori-

entation describing the crack angle in material coordinates is measured from the longitudinal grain direction, which

corresponds to the rolling direction for rolled products. The crack growth resistance is maximum for growth across the roll-

ing direction (h= 90, or LT) and minimum for growth parallel to the rolling direction (0, or TL) (Lemant and Pineau,

1981). For this case of two-dimensional fracture orthotropy, in the context of a maximum stress theory, Buczek and Hera-

kovich (1985)proposed a fracture orthotropy interpolation function by requiring that the toughness function be indepen-

dent ofh for isotropic materials, and that it possess the desired orthogonal symmetry, collocating to Kp (0) and Kp (90)

values, and is given by

Kph Kp0 cos2 hKp90 sin2 h 14

Fig. 11. Mode I SIF (KI) for cases 0 and 14; crack inclination / = 0. [a= 0.254 m (1 in.),r = 6895 Pa (1 psi)].



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Fig. 12. Mode I SIF (KI) for cases 0 and 14; crack inclination / = 30. [a= 0.254 m (1 in.),r = 6895 Pa (1psi)].

Fig. 13. Mode II SIF (KII) for cases 0 and 14; crack inclination / = 30. [a= 0.254 m (1 in),r = 6895 Pa (1psi)].



11/16

whereKpis taken to represent the stress intensity at which the crack propagates and is considered to be a material-depen-

dent function of the orientation of the crack tip consistent with the regime of crack growth. Kfouri (1996) proposed the ellip-

tical form shown below:

1

Kph2 cos

2 h

Kp02 sin

2h

Kp902 15

Either case produces a nearly identical oblong shape in polar coordinates for fairly small orthotropy ratios, as illustrated in

Fig. 15(Pettit, 2000).

Extension of 2D fracture orthotropy relations to 3D is not straightforward, nor is there an established or accepted

method to do so.Pettit (2000)accounts for three-dimensional orthotropy by defining a function that interpolates the frac-

ture resistance for arbitrary orientations in terms of six principal fracture toughness values, and we have adapted his ap-

proach to the foam material. In a three-dimensional body a crack may be non-planar, and oriented arbitrarily. At any point

along the crack front in an orthotropic material, however, we can characterize the local orientation in terms of the tangent

plane and the crack front normal vector within that plane, defined relative to the principal axes of the material. For an

orthotropic material there are three orthogonal planes of symmetry. Within each of these planes there are thus two

orthogonal axes of symmetry. This results in six principal fracture toughness values. The material is assumed to be homo-

geneous, thus the toughness for a given orientation relative to these principal planes is invariant with regard to transla-

tion. Following the convention established for metals (Goode, 1972), the principle values of fracture toughness are written

in a two-letter code (ij) where the first letter refers to the principle axis normal to the crack plane, and the second sub-

script identifies the principle axis corresponding to the direction of propagation. The six principal toughness values are

defined inFig. 16,for a crack rotated arbitrarily relative to the material orientation. The crack (or a point on an arbitrary

crack front) may propagate in an arbitrary direction defined by unit vector a= a1i+ a2 j+ a3k, where i, j, and kare unit

vectors corresponding to the principal material axes. Vector a lies within a plane tangent to the developing crack surface

at the crack front, which plane is uniquely described by its unit normal vector n = n1i+ n2 j + n3k. The crack orientation is

uniquely defined by the direction cosines aiandni. Following the work ofBuczek and Herakovich (1985), the interpolation

function must, (i) be independent ofai andni for an isotropic material, and (ii) return the principal fracture resistances for

cracks in the corresponding principal orientations. Seeking the lowest order function that can achieve this that also reverts

to the two-dimensional form of Eq. (14), Pettit (2000) derives the fracture resistance components of a

in the principal

planes as

Fig. 14. Mode III SIF (KIII) for cases 0 and 14; crack inclination / = 30. [a= 0.254 m (1 in.),r = 6895 Pa (1 psi)].



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K1a 11 a21

K12a22K13a23

K2a 1

1 a22K23a

23K21a21

K3a 1

1a23

K31a21K32a22

16

These components are summed in some weighted combination based on crack-plane normal n to obtain the effective frac-

ture resistance, Kp(a,n) orKeff(a, n). Since the weight factors must sum to unity to satisfy the isotropic case, Pettit (2000)

expresses the effective fracture resistance as

Kpa;n Keffa;n K1n21K2n22K3n23 n

21

1 a21K12a

22K13a23

n221 a22

K23a23K21a21

n231 a23

K31a21K32a22

17Using the mode I toughness values measured in the three principal directions K1223 psi


p 25; 057 Pa ffiffiffiffiffimp ;

K1317:4 psiffiffiffiffiffiffi

in:p

19; 122 Pa ffiffiffiffiffimp ; K2119:5 psi ffiffiffiffiffiffiin:p 21; 430 Pa ffiffiffiffiffimp , we use Eq.(17)to estimate toughness in an arbitrarydirection. For the transversely isotropic material we have three unique toughness values and are taking K23= K13,K31= K21,

andK12= K32. We note that we are only making use of the mode I toughness values measured in the three principal 11, 22

and 33 directions. We have omitted the use ofKII

andKIII

since there is no accepted three-dimensional fracture criterion for

anisotropic material under mixed-mode loading conditions. For arbitrary crack angles and material orientations, KIIandKIII

Kp(0 )

Kp(90)

r

r= Kp()

Fig. 15. Assumed elliptical function describing two-dimensional crack growth resistance as a function of orientation for materials with fracture orthotropy.

Fig. 16. Anisotropic toughness [Pettit, 2000].



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are expected to play a role in determining both Keffand crack turning angle. The fracture criterion based on Eq.(17)is a first

attempt at evaluating fracture properties of anisotropic foam, a material of engineering importance. Efforts should be made

in the future to include mode mixity effects for formulating fracture criterion for anisotropic materials.

7. Crack turning angle

Predicting crack turning angle under mixed-mode loading for anisotropic materials is a complex endeavor involving the

calculation of the stress and displacement fields near the crack tip and accounting for the direction dependence of both theelastic and fracture properties. The three theories prevalent in the literature to predict incipient crack turning angles for iso-

tropic materials are the maximum hoop stress [Erdogan and Sih, 1963], maximum energy release rate (Hussain et al., 1974),

and minimum strain energy density [Sih, 1974].

Buczek and Herakovich (1985)were among the first to predict the crack extension angle for orthotropic composite mate-

rials.Nobile and Carloni (2005) and Carloni et al. (2003)consider incipient crack turning for an orthotropic plate under biax-

ial loading. Mixed-mode crack propagation in anisotropic materials is investigated by Saouma et al. (1987) based on Sih et al.

(1965), under conditions ofz-symmetry.Chen (1999)analyzed crack propagation in aluminum based on a ratio of the hoop

stress to anisotropic fracture toughness, Kc. A similar analysis was conducted by Pettit (2000) for rolled aluminum, which has

isotropic stiffness properties but anisotropic fracture characteristics defined by Eq.(17).Pettit (2000)predicted the aniso-

tropic crack turning angle based on finding the maximum value of the function rhh/Keff (a, n) in the neighborhood of the cracktip, illustrated inFig. 17.

The hoop stress rhh in the neighborhood of the crack tip is calculated based on the general expressions for anisotropic

materials shown below (Hoenig, 1982), and use of appropriate coordinate transformations:

rxxryyryzrzxrxy

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

1ffiffiffiffiffiffiffiffiffi2pr

p

ReP3i1

l2iN1

ij Kj

coshlisin h

ReP3i1

N1ij Kj

coshlisin h

ReP3i1

liN1ij Kj

coshlisin h

ReP3i1

likiN1ij Kj

coshlisin h

ReP

3

i1

kiN1ij Kj

coshlisin h

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

rzz S31rxxS32ryyS34ryzS35rzxS36rxy=S33

18

Fig. 18 shows a fractured M(T) test specimen. The wavy knit lines are marked by dots. The waviness results from difficulty

in spraying in a straight line and also the rise heights differ along the spray line. An average knit line angle nearest to the

crack was used to determine the orientation of the material coordinate axes (x0, y 0, z0), shown inFig. 19. Test results from

x

y a()

n()

Predicatedpropagation path

Keff(a,n)

Fig. 17. Criteria for predicting anisotropic crack turning angle (Pettit, 2000).



14/16

three M(T) specimens, labeled AC, are used to predict crack turning angles. Fabrication of the M(T) specimens involves

spraying the foam on a metal plate, slowing it to rise and form a rind or knit line. When the foam has settled, the next layeris applied until the desired thickness is achieved. From here, the foam is cut from this parent block so the knit lines are run-

ning in the desired directions.

The direction cosines for the material coordinate systems of specimens AC are shown in Table 3. Note that because of the

wavy nature of the knit lines, determination of the direction cosines for the material axes is approximate. The angles wand C

Fig. 18. From left to right: front, left, right, and rear sides of the fractured M(T) specimen.

Fig. 19. Definition of the knit line plane and orientation of material coordinate axes (x0 ,y0 ,z0) The procedure for predicting anisotropic crack turning angle issummarized below (Knudsen, 2006).

Table 3

Direction cosines for the material coordinate axes for specimens AC

Specimen A Specimen B Specimen C

x y z x y z x y z

x0 0.94 0 0.342 0.94 0 0.342 0.77 0 0.64y0 0 1 0 0 0.984 0.174 0.09 0.99

0.1

z0 0.342 0 0.94 0.337 0.163 0.925 0.632 0.133 0.76



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define the knit line plane, as shown in Fig. 19, and /defines the crack inclination angle. The crack inclinations for specimens

AC are 10, 10 and 30. The direction cosines between the (x, y, z) and (x0, y 0, z0) coordinates are obtained by successiverotations about the unprimed and primed axes, to arrive at the final (x0 ,y 0 ,z0) configuration.

The procedure for predicting anisotropic crack turning angle is summarized below ( Knudsen, 2006):

Run the ANSYS FE model with the appropriate material orientation and compute the SIFs (KI,KII,KIII) using FRANC3D. Rotate material properties into a local crack front coordinate system Compute near-tip stresses using Eq.(11)(Hoenig, 1982). Computerhh using near-tip stresses and appropriate coordinate transformations. Determine the angle hat which the function rhh/Keff(a, n) reaches a maximum value, to predict the crack turning angle, hc.

Table 4 presents measured and predicted crack turning angles for specimens AC. Turning angles based on isotropicassumptions are also presented. The SIFs can vary along the crack front and since the free surface can influence the numerical

K-solution, the measured and predicted angles are shown for the specimen mid-plane. The turning angle was measured

using a Brown and Sharpe coordinate measuring machine. Specimen A has the best agreement between measured and pre-

dicted results. Errors can result from the measurement of the material coordinate axes. When defining the knit line plane, the

knit line closest to the crack was used to define the orientation, even though the material orientation can vary over the spec-

imen. For the three specimens tested, the crack tends to turn in the mode I direction, perpendicular to the max principal

stress direction. More testing is required to generalize results.

The Keffexpression used (Eq.(17)) does not include the effects of mode mixity arising from KIIand KIII. Inclusion of effects

of mixed-mode loading may result in better agreement between experimental and calculated results for the crack propaga-

tion angle.

8. Conclusions

Comprehensive results for anisotropic mixed-mode SIFs (KI, KII, KIII) for BX-265 foam material, used for insulating the

shuttle cryogenic external tank, are presented. Based on experimental test data, this foam material is modeled as trans-

versely isotropic linear elastic solid. We present a general procedure for computing mixed-mode SIFs for anisotropic mate-

rials. Middle-tension M(T) finite element models with 17 different material orientations were analyzed and the mixed-mode

SIFs (KI,KII,KIII) were computed using FRANC3D software developed by the Cornell University Fracture Group. The SIFs are

presented as a function of specimen through thickness, and crack inclination, for 17 material orientations. We observe that

even when the material primary axis was offset by 45,KIvaried by only 14%, indicating that toughness is not a strong func-

tion of material orientation. We present fracture toughness values for the foam material in three primary directions and also

a means for evaluating directional dependence of anisotropic toughness based on work by Pettit (2000). A detailed procedure

for predicting anisotropic crack turning angles is presented and theoretical predictions are compared with measured values.

The results presented, based on comprehensive numerical and experimental investigations, represent a quantitative basis

for evaluating the strength and fracture properties of BX-265 anisotropic foam material, used to insulate the space shuttle

external tank. Inclusion of mixed-mode loading effects for determining Kefffor anisotropic materials is recommended for fu-

ture work.

Acknowledgments

This work was supported by the NASA Marshall Space Flight Center, Huntsville, AL. Discussions on theoretical aspects and

finite element modeling of cracks in anisotropic materials with Drs. B. Carter and P. Wawrzynek of the Cornell University

Fracture Group, and Dr. R. Pettit at Pratt & Whitney, East Hartford, CT, has contributed greatly towards our understanding

and completion of this work.

References

Aliabadi, Rooke, D.P., 1991. Numerical Fracture Mechanics. Computational Mechanics Publications, Boston. 45 pp.

ANSYS Elements Reference, 1999. ANSYS Release 5.6, ANSYS Inc.

Table 4

Measured and predicted crack turning angles

Crack inclination, / hc (measured) hc (predicted) hc(isotropic)

Specimen A 10 27 32 43

Specimen B 10 9.7 21 27

Specimen C 30 12 19 27



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Banks-Sills, L., Hershkovitz, I., Wawrzynek, P.A., Eliasl, R., Ingraffea, A.R., 2005. Methods for calculating stress intensity factors in anisotropic materials. Part

Iz= 0 is a symmetry plane. Engineering Fracture Mechanics 72, 23282358.Banks-Sills, L., Wawrzynek, P.A., Carter, B., Ingraffea, A.R., Hershkovitz, I., 2007. Methods for calculating stress intensity factors in anisotropic materials. Part

II. Arbitrary geometry. Engineering Fracture Mechanics 74, 12931307.

Buczek, M.B., Herakovich, C.T., 1985. A normal stress criterion for crack extension direction in orthotropic composite materials. Journal of Composite

Materials 19, 544553.

Carloni, C., Piva, A., Viola, E., 2003. An alternative complex variable formulation for an inclined crack in an orthotropic medium. Journal of Engineering

Fracture Mechanics 70, 20332058.

Chen, C.S., 1999. Crack growth simulation and residual strength prediction in thin shell structures. Ph.D. Dissertation, Cornell University.

Erdogan, F., Sih, G.C., 1963. On the extension of plates under plane loading and transverse shear. Journal of Basic Engineering 85, 519527.

FRANC3D Concepts and Users Guide, 2003. Available from the Cornell Fracture Group website (http://www.cfg.cornell.edu ), Ithaca.Freed, Y., Banks-Sills, L., 2005. A through interface crack between a 45transversely isotropic pair of materials. International Journal of Fracture 133 (1), 1

44.

Goode, R.J., 1972. Identification of fracture plane orientation, materials research and standards (MIRSA). ASTM 12 (9).

Hoenig, A., 1982. Near-tip behavior of a crack in a plane anisotropic elastic body. Engineering Fracture Mechanics 16 (3), 393403.

Hussain, M.A., Pu, S.L., Underwood, J.H., 1974. Strainenergy release rate for a crack under combined mode I and mode II. Fracture AnalysisASTMSTP 560, 2

28.

Ishikawa, H., 1980. A finite element analysis of stress intensity factors for combined tensile and shear loading by only a virtual crack extension. International

Journal of Fracture 16, 243246.

Kfouri, A.P., 1996. Crack extension under mixed-mode loading in an anisotropic mode-asymmetric material in respect of resistance to fracture. Fatigue &

Fracture of Engineering Materials & Structures 19 (1), 2738.

Knudsen, E.C., 2006. Anisotropic fracture analysis of BX-265 foam insulation material under mixed-mode loading. Ph.D. Dissertation, Mechanical &

Aerospace Engineering Department, University of Florida, Gainesville, FL.

Knudsen, E.C., Arakere, N.K., 2006. Numerical evaluation of Mode I stress intensity factor for BX-265 foam insulation material. Presented at the 51st ASME/

IGTI Turbo Expo International Conference, Barcelona, Spain, May 811, 2006.

Lemant, F., Pineau, A., 1981. Mixed mode fracture of a brittle orthotropic material example of strongly textured zinc sheets. Engineering Fracture

Mechanics 14, 91105.

Li, F.Z., Shih, C.F., Needleman, A., 1985. A comparison of methods for calculating energy release rates. Engineering Fracture Mechanics 21, 405421.Nobile, L., Carloni, C., 2005. Fracture analysis for orthotropic cracked plates. Journal of Composite Structures 68, 285293.

Parks, D.M., 1974. A stiffness derivative finite element technique for the determination of crack tip stress intensity factors. International Journal of Fracture

10 (4).

Pettit, R.G., 2000. Crack turning in integrally stiffened aircraft structures. Ph.D. Dissertation, Cornell University, Ithaca, NY.

Raju, I.S., 1987. Calculation of strain energy release rates with higher order and singular finite elements. Engineering Fracture Mechanics 28 (3), 251274.

Ranjan, S., Arakere, N.K., 2008. A fracturemechanics based methodology for fatigue lifeprediction of single crystal nickel based superalloys. ASME Journal of

Engineering for Gas Turbines and Power 130 (3), 032501, 111.

Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics 35,

379386.

Saouma, V.E., Ayari, M.L., Leavell, D.A., 1987. Mixed mode crack propagation in homogenous anisotropic solids. Journal of Engineering Fracture Mechanics

27, 171184.

Shih, C.F., Moran, B., Nakamura, T., 1986. Energy release rate along a three-dimensional crack front in a thermally stressed body. International Journal of

Fracture 30 (2), 79102.

Sih, G.C., 1974. Strain energy density factor applied to mixed mode crack problems. International Journal of Fracture 10, 321350.

Sih, G.C., Paris, P.C., Irwin, G.R., 1965. On cracks in rectilinearly anisotropic bodies. International Journal of Fracture Mechanics 1, 189202.

Swenson, D., Ingraffea, A., 1987. A finite element model of dynamic crack propagation with an application to intersecting cracks. In: Proceedings of the

Fourth International Conference on Numerical Methods in Fracture Mechanics. San Antonio, TX.Swenson, D., Ingraffea, A., 1988a. Modeling mixed-mode dynamic crack propagation using finite elements: theory and applications. Computational


Swenson, D., Ingraffea, A., 1988b. Using combined experiments and analysis to generate dynamic critical stress intensity data. In: Cruse, T. (Ed.), Fracture

Mechanics: 19th Symposium, vol. 969. ASTM STP, Philadelphia, pp. 405426.

Ting, T.C.T., 1996. Anisotropic Elasticity-Theory and Applications. Oxford University Press. pp. 3437.

Yau, J.F., Wang, S.S., Corten, H.T., 1980. A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. ASME Journal of Applied


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Determination of Mixed-mode Stress Intensity Factors, Fracture